Order 2 (1985), 265-268.
0167~8094/85.15
0 1985 by D. Reidei Pubhshing Company,
265
Largest Induced Suborders Satisfying the Chain
Condition
NATHAN
LINIAL
Department of Computer Science, Hebrew University, Jerusalem, Israel
MICHAEL
SAKS
Bell Communications Research, 43s South St., Morristown,
NJ 07960, U.S.A.
and
PETER SHOR
Department of Mathematics, MIT, Cambridge, MA 02139, U.S.A.
Communkated
by D. Duffis
(Received: 17 June 1985; accepted: 11 September 1985)
Abstract. For a fmite ordered set P, let c(P) denote the
the induced suborder on Q satisfies the Jordan-Dedekind
chain in Q has the same cardinality. For positive integers
ordered setsP of cardinahty n. We prove: &r - 1 <f(n)
AhfS (MOS) subject ckwiflcation
(1980).
cardinality of the largest subset Q such that
chain condition (JDCC), i.e., every maximal
n, let
be the minimum of c(P) over all
< 4e
06AO5.
Key words. Ordered sets, induced suborders, chain condition.
For a finite ordered set P, let c(P) denote the size of the largest subset Q such that Q
with the induced order satisfies the Jordan-Dedekind chain condition (JDCC), i.e., every
maximal chain in Q has the same cardinality. We consider the question, in terms of the
cardinality of P, how small can c(P) be? For positive integers 11,let f(n) be the minimum
of c(P) over all ordered sets P of cardmality K j(n) is trivially at least &, since every
ordered set on n elements has either a chain or antichain of cardinality 6 (a simple
consequence of Dilworth’s theorem [l]). We prove:
THEOREM 1. &
- 1 <j(n)
< 4e&
for aZ1n.
Before giving the proof, let us mention that this is an example of a wide variety of questions of the form: given a ‘nice’ classof ordered sets and given an arbitrary ordered set P,
how large an induced suborder must P have that belongs to this class. Another such question (to which we do not know any nontrivial bounds) arises if we take the classto be the
ordered sets of dimension two. Similar questions have been studied in the context of
graphs and we believe it to be worthwhile to consider such questions for ordered sets.
NATHAN LINIAL
266
ET AL.
The remainder of this paper is devoted to proving Theorem 1. The lower bound of
the theorem is a strengthening of the observation which gave the dn lower bound. Let
A be a largest antichain of P and let B be a largest antichain of P-A. Then every element
of B is related to some element of A (otherwise A is not maximal). Let A’ be the subset
of .4 consisting of all elements that are related to some element of B; 1A’1 > 1B I, otherwise B U (..-A’) is a larger antichain than A. Hence, B U A’ is a height one poset with
no isolated elements (thus satisfying JDCC) having cardinality at least 2 1B 1. If 2 1B I>
6-l
or lAl>fi-l
then c(P)>d%-1
so suppose IA]<&-1
and
1B 1< (A/&$
- i. Then P-A has cardinality greater than n - 6
t 1 and no antichain of size (6/h)
- i, so it has a chain of size (n - 6
+ l)/(&&
- i) > 6
- 1,
which completes the proof of the lower bound.
We now proceed to the upper bound. The proof we present is nonconstructive; we
show that for every n there exists a poset on n elements so that every subposet of size
greater than 4e & does not satisfy JDCC.
Let u be a permutation of [n] (= { 1, 2, . ., n}). The permutation order P(u) associated with u is defmed on the set {xi , . . , X~ } by xi <xi if i < j and u(i) < u(j). (The
class of permutation orders is equivalent to the class of two-dimensional orders and has
been studied extensively.) It will be convenient to view a permutation u as a sequence
41, G9, . . . . u(n). By a subsequence of u we mean a sequence u(il), u(&), . . . . u(ik)
where il < iZ < *.. < ik. Observe that the induced suborders of P(u) are associated to
subsequences of u and are themselves permutation orders. We will say that a permutation
u (or more generally, a subsequence u(il), u(&), . . . , u(ik)) is JDCC if the associated
permutation order (or induced suborder) satisfies JDCC. We let g(n) denote the number
of permutations of [n] satisfying JDCC.
We will prove
THEOREM 2. For any m > 4e&
there exists Q permutation order on n elements
such that any induced suborder of size m or greater fails to satisfy JDCC, and therefore
f(n) < 4eG.
The proof of Theorem 2 is based on two lemmas.
LEMMA 3. Zf for every permutation
m satisfying JDCC then:
LEMMA4.
Forallm,g(m)<
u of n, P(u) has an induced suborder of size at least
16m.
To see that Lemmas 3 and 4 imply Theorem 2, suppose that every permutation order of
size n has an induced suborder satisfying JDCC of size at least m. Then by Lemmas 3 and
4:
LARGEST
INDUCED
SUBORDERS
SATISFYING
THE
CHAIN
Using the inequalities n !/(n - j)! < nj andj! > jj”/ej,
CONDITION
267
we obtain:
If tn > 4e&,
each summand is less than l/m and the sum is less than (m - 1)/m, a
contradiction establishing f( n) < 4e &.
Proof of Lemma 3. We begin with a
Claim. If P has an induced suborder of size at least m satisfying JDCC, then it has such
an induced suborder of size between m and 2m - 2.
To see this, let Q be the smallest induced suborder of P satisfying JDCC and having
size at least m. Assume 1Q I> m; by minimality of Q it is not an antichain. Let jk! be the
set of minimal elements of Q. Then Q #&f and both&f and Q-M satisfy JDCC, so by the
choice of Q, 1M 1< m and ] Q-M ] < m, and so ] Q I< 2m - 2. This establishing the claim.
Now let u be a random permutation of n elements (each permutation is chosen with
probability l/n!). For a given index set iI < iz < ... <ii the probability that the induced
Thus the
suborder corresponding to u(iI ), u(&), . . . , u(ij) satisfies JDCC is g(j)/j!.
probability
that u has a subsequence of size j satisfying JDCC is at most
[I; km/l.!.
If each permutation order of size n has an induced suborder of size at least m satisfying
JDCC, then by the above claim, a random permutation contains a subsequence satisfying
JDCC of size between m and 2m - 2 with probability 1. Hence
of Lemmu 4. To bound g(n), we fast need to characterize permutations satisfying
JDCC. Note that a chain in P(u) corresponds to an increasing subsequence of u. Define
c(i; u) to be the length of the longest increasing subsequence of u ending with u(i), that
is,c(i; u)=max{kl
there existsir <iz < .*.<ik =i such that u(iI)< *.~<u(i~)}.
Proof
Note that an increasing subsequence in u in positions i, < iz < ... < ik corresponds to
an increasing subsequence in u-l in positions u(ir ) < ... < u(ik). In particular, we have
c(i; u) = c(u(i);
u-l).
(*I
The function c(i; u) plays a key role in work of Shensted [4] and Greene [3] concerning
Young tableau and permutations. The following lemma is implicit in their work.
LEMMA 5. Every permutation
u is determined uniquely by the functions c(i; u) and
c(i, u-l).
&oofi Given the functions c(i; u) and c(i; u-i), we show how to reconstruct u. Fix
k < n and let iI < iz < ... <i, (resp. jr < ... < js) be the set of indices i such that
c(i; u) = k (resp. c(i; u-i) = k). By (*), u maps the set {ii , . . ., i,} bijectively to the set
268
NATHAN LINIAL
ET AL.
G r , . . . , js}, so in particular, r = s. Moreover, we must have u(i,,) < u(i,, - I) < . .. < u(iI)
since u(it) < u(it+I) would imply c(it+I; u) > c(it; u). Thus u(iI) = jr, u(i*) =jrpI,
cl
. . . . u(&) =jr . Doing this for all Ic e [n] determines u uniquely.
A function o : [n] + [n] will be said to be udmissible if o(1) = 1 and cx(i) < a(i - 1) + 1
for2GiGn.
LEMMA 6. If u is JDCC then c(i; u) and c(i; u-l) are both admissible.
&oofi Suppose u is JDCC. Clearly ~(1; u)=l. c(i; u)<c(i1; u)+ 1 is trivial if
u(i) < u(i - 1). If u(i) > u(i - 1) then u(i - 1) and u(i) are contained together in some
maximal (hence, maximum) increasing subsequence. The portion of this sequence ending
with u(i) has length c(i; u) (otherwise we could construct a larger maximal chain).
Hence, c(i; u)< c(i - 1; u) + 1 and c(i; u) is admissible. If u is JDCC then so is u-l
(sinceP(u) andP(u-‘) are isomorphic), so c(i; u-r) is admissible.
0
Thus, by Lemmas 5 and 6, g(H) is bounded by the number of pairs o, /I of admissible
functions of [n], i.e., the square of the number of admissible functions, so Lemma 4
follows if we show that there are at most 4n admissible functions of n. Associate to
any function o: [n] -+ [n] the function 7 given by y(i) = 1 + a(i) - cx(i + 1) if 1 <i <
n - 1 and y(n) =a(n). Then ~(1) + ~(2) t .**t T(n) =n, o is determined by 7, and o
is admissible if and only if 7 is nonnegative. Hence, the number of admissible functions
equals the number of nonnegative sequences 7(l), y(2), . . . . T(n) which sum to n. Elementary enumeration yields that there are
, which is less than 4’.
Acknowledgement
The work in this paper was done while the authors were participating in the Discrete
Mathematics Workshop in Budapest, September 1984, sponsored by the Hungarian
Academy of Sciences and the NSF.
References
1. R. P. Dilworth (1950) A decomposition theorem for partially ordered sets, AWZ. M&/I. 51, 161166.
2. P. Erd& and J. Spencer (1974) Probabilistic Methods in Combinatorics, Akad& Kiadbs, Budapest.
3. C. Greene (1974) An extension of Schensted’s Theorem, Adv. Math. 14, 254-265.
4. C. Schensted (1961) Longest increasing and decreasing subsequences, Can. J. Math. 13, 179-191.